Austin Nichols

@austnnchols

Unbiased Instrumental

Variables (IV) in Stata

Magic Bullets

• Instrumental Variables (IV) methods are the only way to estimate causal effects in a variety of settings, including experiments (randomized control trials or RCTs) with imperfect compliance

IV methods often exhibit poor performance

– Bias & size distortion with many weak instruments– No finite moments when exactly identified

• Andrews and Armstrong (2017) offer a solution

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Causal Diagram

• Conditioning on confounders does not in general solve the problem of endogenous participation in a treatment of interest

• The receipt of a treatment (R=1) whose effect b we want to measure may be randomly assigned (Z=1), but we still need IV to estimate impact

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Sign restriction allows unbiased IV

• IV has one fewer moments than overid restrictions, so exactly identified IV has no moments

– Hirano and Porter (2015) show that mean, median, and quantile unbiased estimation are all impossible in the linear IV model with an unrestricted parameter space for the first stage

• This result no longer holds when the sign of the first stage is known (e.g. no defiers, some compliers):

– In models with a single instrumental variable, Andrews and Armstrong (2017) show that there is a unique unbiased estimator based on the reduced form and first-stage regression estimates

– This estimator is substantially less dispersed than the usual 2SLS estimator in finite samples

• In an RCT, we are very confident the first stage is positive

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Model and Estimator

Y=Zpb+u reduced form coef x1=(Z’Z)-1(Z’Y)

R=Zp+v first stage coef x2=(Z’Z)-1(Z’R)

IV estimator constructs Wald ratio x1 / x2

Assume u,v normal so (x1 , x2)~N(m,S) w/variance S=(s12 , s12 \ s12 , s2

2)

Let d=(x1 - x2 s12 /s22). E[d]=pb-ps12 /s2

2

Voinov and Nikulin (1993) show that unbiased estimation of 1/p is possible if its sign is known:

Let t=F( - x2 /s2)/f(x2 /s2)s2 then E[t]= 1/p and E[dt]= E[d]E[t]= b-s12 /s22

Estimator bU=dt+s12/v2

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Further considerations

• bU is asymptotically equivalent to 2SLS when instruments are strong and thus bU can be used together with conventional 2SLS standard errors

• Optimal estimation and optimal testing are distinct questions in the context of weak instruments

– bU is uniformly minimum risk unbiased for convex loss, but it follows from the results of Moreira (2009) that the Anderson–Rubin test is the uniformly most powerful unbiased two-sided test in the just-identified context (not a conditional t-test based on bU)

– more research needed on tests based on this unbiased IV estimator…

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Small-Sample Properties

• Note this applies to bivariate normal errors with known variance, not the focal case of random assignment Z={0,1} and endogenous receipt of treatment R={0,1}

– Appendix B (Nonnormal errors and unknown reduced-form variance) “derives asymptotic results for the case with non-normal errors and an estimated reduced-form covariance matrix. Appendix B.1 shows asymptotic unbiasedness in the weak-instrument case. Appendix B.2 shows asymptotic equivalence with 2SLS in the strong-instrument case”

– How does this approach perform in finite samples?

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Stata command

• Estimator implemented as aaniv on SSC

• Download using ssc install aaniv

• So far, just one endogenous treatment and one

excluded instrument (as of today), as is ideal for

an RCT, but the command will be updated in

future releases to a larger set of use cases

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Small-Sample Properties

• Even with binary R and Z, so non-normal errors by design, standard linear regression rejects the truth all the time, and unbiased IV outperforms standard IV/2SLS

(this simulation has a high correlation between a normal variate that predicts R and the unobserved error that predicts the outcome Y)

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Distributions of Estimators

by Sample Size and Correlation

Sample sizes Correlation of u,v

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Rejection rates about right for IV models,

in large samples

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Conclusion

• Unbiased IV performs as well as IV-2SLS in a setting that it is not designed for, with no bias and lower evident dispersion (but neither has a finite variance)

– Report unbiased IV for an experiment, if only to enable meta-analysis; use aaniv (ssc install aaniv) in Stata

• Rejection rates for both Unbiased IV and IV 2SLS approximately at the nominal rate when sample size is over a thousand

– At smaller sample sizes, there is some under-rejection of a true null—needs further study

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abtassociates.com

ContactAustin Nichols

Principal Scientist

austinnichols@gmail.com